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Equivalence of finite-selection games for ω-covers and Vietoris finite-range open covers

Establish whether, for every topological space X, the finite-selection games G_fin(Ω_X, Ω_X) and G_fin(𝒪_{𝔽(X,ord)}, 𝒪_{𝔽(X,ord)}) are equivalent in the sense of mutual II⁺-reductions; that is, determine whether G_fin(Ω_X, Ω_X) ↔ G_fin(𝒪_{𝔽(X,ord)}, 𝒪_{𝔽(X,ord)}) holds universally.

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Background

The paper compares selection games tied to covering properties of a ground space X and to its Vietoris-power-derived spaces. After showing several non-transfer phenomena in the single-selection setting, the authors note that σ-compact examples do not witness failures for finite selections and state they have not identified any counterexamples. They then pose the equivalence question for the finite-selection games, which would tightly link ω-Menger behavior of X and Menger behavior of the finite-range Vietoris power 𝔽(X,ord).

References

In fact, we have yet to identify any examples of a space extending Example \ref{example:OmegaRothbergerNontransfer} to finite-selections. Is it possible that, for spaces X, \mathsf G_{\mathrm{fin}(\Omega_X, \Omega_X) \leftrightarrows \mathsf G_{\mathrm{fin}(\mathcal O_{\mathbb F(X, \mathrm{ord})},\mathcal O_{\mathbb F(X, \mathrm{ord})})?

— An Adaptation of the Vietoris Topology for Ordered Compact Sets (2507.17936 - Caruvana et al., 23 Jul 2025) in Question 5.? (labelled Question \ref{question:MengerThing}), Subsection “Commentary on Covering Games”